Two metal wires of identical dimensions are connected in series. If σ$$1$$ and σ$$2$$ are the conductivities of the metal wires respectively, the effective conductivity of the combination is

Question

Two metal wires of identical dimensions are connected in series. If σ$$1$$ and σ$$2$$  are the conductivities of the metal wires respectively, the effective conductivity of the combination is

Infinity Learn · Accepted Answer

As both metal wires are of identical dimensions, so their length and area of $$cross-section$$ will be same. Let them be I and A respectively. Then The resistance of the first wire $$isR1=l$$σ$$1A$$                                ....(i)and$$ that of the second wire is $$R2=l$$σ$$2A$$                       ...$$(ii)As$$ they are connected in series, so their effective resistance $$isRs=R1+R2=l$$σ$$1A+l$$σ$$2A$$  $$($$ using $$(i) and  (ii))=lA1$$σ$$1+1$$σ$$2$$                  . . .$$(iii)If$$ σeff is the effective conductivity of the combination, $$thenRs=2l$$σeff A                        . . . .$$(iv)Equating$$ eqns. $$(iii) and (iv), we get $$2l$$σ$$effA=lA1$$σ$$1+1$$σ$$22$$σeff $$=$$σ$$2+$$σ$$1$$σ$$1$$σ$$2$$⇒σeff $$=2$$σ$$1$$σ$$2$$σ$$1+$$σ$$2$$

Two metal wires of identical dimensions are connected in series. If $σ_{1} a n d σ_{2}$ are the conductivities of the metal wires respectively, the effective conductivity of the combination is

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Two metal wires of identical dimensions are connected in series. If σ1 and σ2 are the conductivities of the metal wires respectively, the effective conductivity of the combination is

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Two metal wires of identical dimensions are connected in series. If $σ_{1} a n d σ_{2}$ are the conductivities of the metal wires respectively, the effective conductivity of the combination is